Theorem clwwlknun 16682

Description: The set of closed walks of fixed length N in a simple graph G is the union of the closed walks of the fixed length N on each of the vertices of graph G. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 3-Mar-2022.) (Proof shortened by AV, 28-Mar-2022.)

Hypothesis

Ref	Expression
clwwlknun.v	|- V = (Vtx ` G )

Assertion

Ref	Expression
clwwlknun	|- ( G e. USGraph -> ( N ClWWalksN G ) = U_ x e. V ( x (ClWWalksNOn ` G ) N ) )

Distinct variable groups:   x, G   x, N   x, V

Proof of Theorem clwwlknun

Dummy variables y i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 4016	. . 3 |- ( y e. U_ x e. V ( x (ClWWalksNOn ` G ) N ) <-> E. x e. V y e. ( x (ClWWalksNOn ` G ) N ) )
2		isclwwlknon 16671	. . . . 5 |- ( y e. ( x (ClWWalksNOn ` G ) N ) <-> ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) )
3	2	rexbii 2557	. . . 4 |- ( E. x e. V y e. ( x (ClWWalksNOn ` G ) N ) <-> E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) )
4		simpl 109	. . . . . 6 |- ( ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) -> y e. ( N ClWWalksN G ) )
5	4	rexlimivw 2664	. . . . 5 |- ( E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) -> y e. ( N ClWWalksN G ) )
6		clwwlknun.v	. . . . . . . . 9 |- V = (Vtx ` G )
7		eqid 2238	. . . . . . . . 9 |- (Edg ` G ) = (Edg ` G )
8	6, 7	clwwlknp 16658	. . . . . . . 8 |- ( y e. ( N ClWWalksN G ) -> ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) )
9	8	anim2i 342	. . . . . . 7 |- ( ( G e. USGraph /\ y e. ( N ClWWalksN G ) ) -> ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) )
10	7, 6	usgrpredgv 16439	. . . . . . . . . . . . 13 |- ( ( G e. USGraph /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) -> ( (lastS ` y ) e. V /\ ( y ` 0 ) e. V ) )
11	10	ex 115	. . . . . . . . . . . 12 |- ( G e. USGraph -> ( { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) -> ( (lastS ` y ) e. V /\ ( y ` 0 ) e. V ) ) )
12		simpr 110	. . . . . . . . . . . 12 |- ( ( (lastS ` y ) e. V /\ ( y ` 0 ) e. V ) -> ( y ` 0 ) e. V )
13	11, 12	syl6com 35	. . . . . . . . . . 11 |- ( { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) -> ( G e. USGraph -> ( y ` 0 ) e. V ) )
14	13	3ad2ant3 1051	. . . . . . . . . 10 |- ( ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) -> ( G e. USGraph -> ( y ` 0 ) e. V ) )
15	14	impcom 125	. . . . . . . . 9 |- ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) -> ( y ` 0 ) e. V )
16		simpr 110	. . . . . . . . . . . 12 |- ( ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) /\ x = ( y ` 0 ) ) -> x = ( y ` 0 ) )
17	16	eqcomd 2244	. . . . . . . . . . 11 |- ( ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) /\ x = ( y ` 0 ) ) -> ( y ` 0 ) = x )
18	17	biantrud 304	. . . . . . . . . 10 |- ( ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) /\ x = ( y ` 0 ) ) -> ( y e. ( N ClWWalksN G ) <-> ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) ) )
19	18	bicomd 141	. . . . . . . . 9 |- ( ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) /\ x = ( y ` 0 ) ) -> ( ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) <-> y e. ( N ClWWalksN G ) ) )
20	15, 19	rspcedv 2933	. . . . . . . 8 |- ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) -> ( y e. ( N ClWWalksN G ) -> E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) ) )
21	20	adantld 278	. . . . . . 7 |- ( ( G e. USGraph /\ ( ( y e. Word V /\ (♯ ` y ) = N ) /\ A. i e. ( 0..^ ( N - 1 ) ) { ( y ` i ) , ( y ` ( i + 1 ) ) } e. (Edg ` G ) /\ { (lastS ` y ) , ( y ` 0 ) } e. (Edg ` G ) ) ) -> ( ( G e. USGraph /\ y e. ( N ClWWalksN G ) ) -> E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) ) )
22	9, 21	mpcom 36	. . . . . 6 |- ( ( G e. USGraph /\ y e. ( N ClWWalksN G ) ) -> E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) )
23	22	ex 115	. . . . 5 |- ( G e. USGraph -> ( y e. ( N ClWWalksN G ) -> E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) ) )
24	5, 23	impbid2 143	. . . 4 |- ( G e. USGraph -> ( E. x e. V ( y e. ( N ClWWalksN G ) /\ ( y ` 0 ) = x ) <-> y e. ( N ClWWalksN G ) ) )
25	3, 24	bitrid 192	. . 3 |- ( G e. USGraph -> ( E. x e. V y e. ( x (ClWWalksNOn ` G ) N ) <-> y e. ( N ClWWalksN G ) ) )
26	1, 25	bitr2id 193	. 2 |- ( G e. USGraph -> ( y e. ( N ClWWalksN G ) <-> y e. U_ x e. V ( x (ClWWalksNOn ` G ) N ) ) )
27	26	eqrdv 2236	1 |- ( G e. USGraph -> ( N ClWWalksN G ) = U_ x e. V ( x (ClWWalksNOn ` G ) N ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   E.wrex 2529   {cpr 3710   U_ciun 4012   ` cfv 5377  (class class class)co 6085   0cc0 8179   1c1 8180   + caddc 8182   - cmin 8497  ..^cfzo 10549  ♯chash 11214  Word cword 11304  lastSclsw 11349  Vtxcvtx 16253  Edgcedg 16298  USGraphcusgr 16395   ClWWalksN cclwwlkn 16644  ClWWalksNOncclwwlknon 16667

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-map 6924  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-dec 9778  df-uz 9922  df-fz 10412  df-fzo 10550  df-ihash 11215  df-word 11305  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256  df-edg 16299  df-umgren 16335  df-usgren 16397  df-clwwlk 16633  df-clwwlkn 16645  df-clwwlknon 16668

This theorem is used by: (None)